The average of a, b, and c is 20, and the average of b, c, and d is 25. If d = 30, what is the value of a?

Introduction:
Two overlapping triples with known averages allow quick recovery of shared pair sums and then the remaining unknown. Translate averages to totals, subtract, and isolate a.

Given Data / Assumptions:
(a + b + c) / 3 = 20 ⇒ a + b + c = 60. (b + c + d) / 3 = 25 ⇒ b + c + d = 75. d = 30.

Concept / Approach:
First compute b + c from the second total using d, then substitute into the first total to solve for a.

Step-by-Step Solution:
b + c = 75 − d = 75 − 30 = 45. a = (a + b + c) − (b + c) = 60 − 45 = 15.

Verification / Alternative check:
Plugging back: a + b + c = 15 + 45 = 60 (OK). b + c + d = 45 + 30 = 75 (OK).

Why Other Options Are Wrong:
25, 30, 45, 20: Do not satisfy both given average relationships when d = 30.

Common Pitfalls:
Averaging the two averages directly, or treating a as equal to d. Always compute pair sums first from totals.

Final Answer:
The value of a is 15.